Calculus Examples

$y = \frac{3}{2} \cdot \sin (x + \frac{π}{3})$

Step 1

Write $y = \frac{3}{2} \cdot \sin (x + \frac{π}{3})$ as a function.

$f (x) = \frac{3}{2} \cdot \sin (x + \frac{π}{3})$

Step 2

Find the first derivative of the function.

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Step 2.1

Since $\frac{3}{2}$ is constant with respect to $x$ , the derivative of $\frac{3}{2} \cdot \sin (x + \frac{π}{3})$ with respect to $x$ is $\frac{3}{2} \frac{d}{d x} [\sin (x + \frac{π}{3})]$ .

$\frac{3}{2} \frac{d}{d x} [\sin (x + \frac{π}{3})]$

Step 2.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \sin (x)$ and $g (x) = x + \frac{π}{3}$ .

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Step 2.2.1

To apply the Chain Rule, set $u$ as $x + \frac{π}{3}$ .

$\frac{3}{2} (\frac{d}{d u} [\sin (u)] \frac{d}{d x} [x + \frac{π}{3}])$

Step 2.2.2

The derivative of $\sin (u)$ with respect to $u$ is $\cos (u)$ .

$\frac{3}{2} (\cos (u) \frac{d}{d x} [x + \frac{π}{3}])$

Step 2.2.3

Replace all occurrences of $u$ with $x + \frac{π}{3}$ .

$\frac{3}{2} (\cos (x + \frac{π}{3}) \frac{d}{d x} [x + \frac{π}{3}])$

Step 2.3

Differentiate.

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Step 2.3.1

Combine $\cos (x + \frac{π}{3})$ and $\frac{3}{2}$ .

$\frac{\cos (x + \frac{π}{3}) \cdot 3}{2} \frac{d}{d x} [x + \frac{π}{3}]$

Step 2.3.2

Move $3$ to the left of $\cos (x + \frac{π}{3})$ .

$\frac{3 \cdot \cos (x + \frac{π}{3})}{2} \frac{d}{d x} [x + \frac{π}{3}]$

Step 2.3.3

By the Sum Rule, the derivative of $x + \frac{π}{3}$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [\frac{π}{3}]$ .

$\frac{3 \cos (x + \frac{π}{3})}{2} (\frac{d}{d x} [x] + \frac{d}{d x} [\frac{π}{3}])$

Step 2.3.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$\frac{3 \cos (x + \frac{π}{3})}{2} (1 + \frac{d}{d x} [\frac{π}{3}])$

Step 2.3.5

Since $\frac{π}{3}$ is constant with respect to $x$ , the derivative of $\frac{π}{3}$ with respect to $x$ is $0$ .

$\frac{3 \cos (x + \frac{π}{3})}{2} (1 + 0)$

Step 2.3.6

Simplify the expression.

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Step 2.3.6.1

Add $1$ and $0$ .

$\frac{3 \cos (x + \frac{π}{3})}{2} \cdot 1$

Step 2.3.6.2

Multiply $\frac{3 \cos (x + \frac{π}{3})}{2}$ by $1$ .

$\frac{3 \cos (x + \frac{π}{3})}{2}$

Step 3

Find the second derivative of the function.

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Step 3.1

Since $\frac{3}{2}$ is constant with respect to $x$ , the derivative of $\frac{3 \cos (x + \frac{π}{3})}{2}$ with respect to $x$ is $\frac{3}{2} \frac{d}{d x} [\cos (x + \frac{π}{3})]$ .

$f'' (x) = \frac{3}{2} \cdot \frac{d}{d x} (\cos (x + \frac{π}{3}))$

Step 3.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \cos (x)$ and $g (x) = x + \frac{π}{3}$ .

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Step 3.2.1

To apply the Chain Rule, set $u$ as $x + \frac{π}{3}$ .

$f'' (x) = \frac{3}{2} \cdot (\frac{d}{d u} (\cos (u)) \frac{d}{d x} (x + \frac{π}{3}))$

Step 3.2.2

The derivative of $\cos (u)$ with respect to $u$ is $- \sin (u)$ .

$f'' (x) = \frac{3}{2} \cdot (- \sin (u) \frac{d}{d x} (x + \frac{π}{3}))$

Step 3.2.3

Replace all occurrences of $u$ with $x + \frac{π}{3}$ .

$f'' (x) = \frac{3}{2} \cdot (- \sin (x + \frac{π}{3}) \frac{d}{d x} (x + \frac{π}{3}))$

Step 3.3

Differentiate.

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Step 3.3.1

Combine $\frac{3}{2}$ and $\sin (x + \frac{π}{3})$ .

$f'' (x) = - \frac{3 \sin (x + \frac{π}{3})}{2} \cdot \frac{d}{d x} (x + \frac{π}{3})$

Step 3.3.2

By the Sum Rule, the derivative of $x + \frac{π}{3}$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [\frac{π}{3}]$ .

$f'' (x) = - \frac{3 \sin (x + \frac{π}{3})}{2} \cdot (\frac{d}{d x} (x) + \frac{d}{d x} (\frac{π}{3}))$

Step 3.3.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$f'' (x) = - \frac{3 \sin (x + \frac{π}{3})}{2} \cdot (1 + \frac{d}{d x} (\frac{π}{3}))$

Step 3.3.4

Since $\frac{π}{3}$ is constant with respect to $x$ , the derivative of $\frac{π}{3}$ with respect to $x$ is $0$ .

$f'' (x) = - \frac{3 \sin (x + \frac{π}{3})}{2} \cdot (1 + 0)$

Step 3.3.5

Simplify the expression.

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Step 3.3.5.1

Add $1$ and $0$ .

$f'' (x) = - \frac{3 \sin (x + \frac{π}{3})}{2} \cdot 1$

Step 3.3.5.2

Multiply $- 1$ by $1$ .

$f'' (x) = - \frac{3 \sin (x + \frac{π}{3})}{2}$

Step 4

To find the local maximum and minimum values of the function, set the derivative equal to $0$ and solve.

$\frac{3 \cos (x + \frac{π}{3})}{2} = 0$

Step 5

Set the numerator equal to zero.

$3 \cos (x + \frac{π}{3}) = 0$

Step 6

Solve the equation for $x$ .

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Step 6.1

Divide each term in $3 \cos (x + \frac{π}{3}) = 0$ by $3$ and simplify.

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Step 6.1.1

Divide each term in $3 \cos (x + \frac{π}{3}) = 0$ by $3$ .

$\frac{3 \cos (x + \frac{π}{3})}{3} = \frac{0}{3}$

Step 6.1.2

Simplify the left side.

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Step 6.1.2.1

Cancel the common factor of $3$ .

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Step 6.1.2.1.1

Cancel the common factor.

$\frac{3 \cos (x + \frac{π}{3})}{3} = \frac{0}{3}$

Step 6.1.2.1.2

Divide $\cos (x + \frac{π}{3})$ by $1$ .

$\cos (x + \frac{π}{3}) = \frac{0}{3}$

Step 6.1.3

Simplify the right side.

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Step 6.1.3.1

Divide $0$ by $3$ .

$\cos (x + \frac{π}{3}) = 0$

Step 6.2

Take the inverse cosine of both sides of the equation to extract $x$ from inside the cosine.

$x + \frac{π}{3} = \arccos (0)$

Step 6.3

Simplify the right side.

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Step 6.3.1

The exact value of $\arccos (0)$ is $\frac{π}{2}$ .

$x + \frac{π}{3} = \frac{π}{2}$

Step 6.4

Move all terms not containing $x$ to the right side of the equation.

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Step 6.4.1

Subtract $\frac{π}{3}$ from both sides of the equation.

$x = \frac{π}{2} - \frac{π}{3}$

Step 6.4.2

To write $\frac{π}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$x = \frac{π}{2} \cdot \frac{3}{3} - \frac{π}{3}$

Step 6.4.3

To write $- \frac{π}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$x = \frac{π}{2} \cdot \frac{3}{3} - \frac{π}{3} \cdot \frac{2}{2}$

Step 6.4.4

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 6.4.4.1

Multiply $\frac{π}{2}$ by $\frac{3}{3}$ .

$x = \frac{π \cdot 3}{2 \cdot 3} - \frac{π}{3} \cdot \frac{2}{2}$

Step 6.4.4.2

Multiply $2$ by $3$ .

$x = \frac{π \cdot 3}{6} - \frac{π}{3} \cdot \frac{2}{2}$

Step 6.4.4.3

Multiply $\frac{π}{3}$ by $\frac{2}{2}$ .

$x = \frac{π \cdot 3}{6} - \frac{π \cdot 2}{3 \cdot 2}$

Step 6.4.4.4

Multiply $3$ by $2$ .

$x = \frac{π \cdot 3}{6} - \frac{π \cdot 2}{6}$

Step 6.4.5

Combine the numerators over the common denominator.

$x = \frac{π \cdot 3 - π \cdot 2}{6}$

Step 6.4.6

Simplify the numerator.

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Step 6.4.6.1

Move $3$ to the left of $π$ .

$x = \frac{3 \cdot π - π \cdot 2}{6}$

Step 6.4.6.2

Multiply $2$ by $- 1$ .

$x = \frac{3 π - 2 π}{6}$

Step 6.4.6.3

Subtract $2 π$ from $3 π$ .

$x = \frac{π}{6}$

Step 6.5

The cosine function is positive in the first and fourth quadrants. To find the second solution, subtract the reference angle from $2 π$ to find the solution in the fourth quadrant.

$x + \frac{π}{3} = 2 π - \frac{π}{2}$

Step 6.6

Solve for $x$ .

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Step 6.6.1

Simplify $2 π - \frac{π}{2}$ .

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Step 6.6.1.1

To write $2 π$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$x + \frac{π}{3} = 2 π \cdot \frac{2}{2} - \frac{π}{2}$

Step 6.6.1.2

Combine fractions.

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Step 6.6.1.2.1

Combine $2 π$ and $\frac{2}{2}$ .

$x + \frac{π}{3} = \frac{2 π \cdot 2}{2} - \frac{π}{2}$

Step 6.6.1.2.2

Combine the numerators over the common denominator.

$x + \frac{π}{3} = \frac{2 π \cdot 2 - π}{2}$

Step 6.6.1.3

Simplify the numerator.

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Step 6.6.1.3.1

Multiply $2$ by $2$ .

$x + \frac{π}{3} = \frac{4 π - π}{2}$

Step 6.6.1.3.2

Subtract $π$ from $4 π$ .

$x + \frac{π}{3} = \frac{3 π}{2}$

Step 6.6.2

Move all terms not containing $x$ to the right side of the equation.

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Step 6.6.2.1

Subtract $\frac{π}{3}$ from both sides of the equation.

$x = \frac{3 π}{2} - \frac{π}{3}$

Step 6.6.2.2

To write $\frac{3 π}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$x = \frac{3 π}{2} \cdot \frac{3}{3} - \frac{π}{3}$

Step 6.6.2.3

To write $- \frac{π}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$x = \frac{3 π}{2} \cdot \frac{3}{3} - \frac{π}{3} \cdot \frac{2}{2}$

Step 6.6.2.4

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 6.6.2.4.1

Multiply $\frac{3 π}{2}$ by $\frac{3}{3}$ .

$x = \frac{3 π \cdot 3}{2 \cdot 3} - \frac{π}{3} \cdot \frac{2}{2}$

Step 6.6.2.4.2

Multiply $2$ by $3$ .

$x = \frac{3 π \cdot 3}{6} - \frac{π}{3} \cdot \frac{2}{2}$

Step 6.6.2.4.3

Multiply $\frac{π}{3}$ by $\frac{2}{2}$ .

$x = \frac{3 π \cdot 3}{6} - \frac{π \cdot 2}{3 \cdot 2}$

Step 6.6.2.4.4

Multiply $3$ by $2$ .

$x = \frac{3 π \cdot 3}{6} - \frac{π \cdot 2}{6}$

Step 6.6.2.5

Combine the numerators over the common denominator.

$x = \frac{3 π \cdot 3 - π \cdot 2}{6}$

Step 6.6.2.6

Simplify the numerator.

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Step 6.6.2.6.1

Multiply $3$ by $3$ .

$x = \frac{9 π - π \cdot 2}{6}$

Step 6.6.2.6.2

Multiply $2$ by $- 1$ .

$x = \frac{9 π - 2 π}{6}$

Step 6.6.2.6.3

Subtract $2 π$ from $9 π$ .

$x = \frac{7 π}{6}$

Step 6.7

The solution to the equation $x + \frac{π}{3} = \frac{π}{2}$ .

$x = \frac{π}{6}, \frac{7 π}{6}$

Step 7

Evaluate the second derivative at $x = \frac{π}{6}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$- \frac{3 \sin ((\frac{π}{6}) + \frac{π}{3})}{2}$

Step 8

Evaluate the second derivative.

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Step 8.1

Simplify the numerator.

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Step 8.1.1

To write $\frac{π}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$- \frac{3 \sin (\frac{π}{6} + \frac{π}{3} \cdot \frac{2}{2})}{2}$

Step 8.1.2

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 8.1.2.1

Multiply $\frac{π}{3}$ by $\frac{2}{2}$ .

$- \frac{3 \sin (\frac{π}{6} + \frac{π \cdot 2}{3 \cdot 2})}{2}$

Step 8.1.2.2

Multiply $3$ by $2$ .

$- \frac{3 \sin (\frac{π}{6} + \frac{π \cdot 2}{6})}{2}$

Step 8.1.3

Combine the numerators over the common denominator.

$- \frac{3 \sin (\frac{π + π \cdot 2}{6})}{2}$

Step 8.1.4

Simplify the numerator.

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Step 8.1.4.1

Move $2$ to the left of $π$ .

$- \frac{3 \sin (\frac{π + 2 \cdot π}{6})}{2}$

Step 8.1.4.2

Add $π$ and $2 π$ .

$- \frac{3 \sin (\frac{3 π}{6})}{2}$

Step 8.1.5

Cancel the common factor of $3$ and $6$ .

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Step 8.1.5.1

Factor $3$ out of $3 π$ .

$- \frac{3 \sin (\frac{3 (π)}{6})}{2}$

Step 8.1.5.2

Cancel the common factors.

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Step 8.1.5.2.1

Factor $3$ out of $6$ .

$- \frac{3 \sin (\frac{3 π}{3 \cdot 2})}{2}$

Step 8.1.5.2.2

Cancel the common factor.

$- \frac{3 \sin (\frac{3 π}{3 \cdot 2})}{2}$

Step 8.1.5.2.3

Rewrite the expression.

$- \frac{3 \sin (\frac{π}{2})}{2}$

Step 8.1.6

The exact value of $\sin (\frac{π}{2})$ is $1$ .

$- \frac{3 \cdot 1}{2}$

Step 8.2

Multiply $3$ by $1$ .

$- \frac{3}{2}$

Step 9

$x = \frac{π}{6}$ is a local maximum because the value of the second derivative is negative. This is referred to as the second derivative test.

$x = \frac{π}{6}$ is a local maximum

Step 10

Find the y-value when $x = \frac{π}{6}$ .

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Step 10.1

Replace the variable $x$ with $\frac{π}{6}$ in the expression.

$f (\frac{π}{6}) = \frac{3}{2} \cdot \sin ((\frac{π}{6}) + \frac{π}{3})$

Step 10.2

Simplify the result.

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Step 10.2.1

To write $\frac{π}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f (\frac{π}{6}) = \frac{3}{2} \cdot \sin (\frac{π}{6} + \frac{π}{3} \cdot \frac{2}{2})$

Step 10.2.2

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 10.2.2.1

Multiply $\frac{π}{3}$ by $\frac{2}{2}$ .

$f (\frac{π}{6}) = \frac{3}{2} \cdot \sin (\frac{π}{6} + \frac{π \cdot 2}{3 \cdot 2})$

Step 10.2.2.2

Multiply $3$ by $2$ .

$f (\frac{π}{6}) = \frac{3}{2} \cdot \sin (\frac{π}{6} + \frac{π \cdot 2}{6})$

Step 10.2.3

Combine the numerators over the common denominator.

$f (\frac{π}{6}) = \frac{3}{2} \cdot \sin (\frac{π + π \cdot 2}{6})$

Step 10.2.4

Simplify the numerator.

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Step 10.2.4.1

Move $2$ to the left of $π$ .

$f (\frac{π}{6}) = \frac{3}{2} \cdot \sin (\frac{π + 2 \cdot π}{6})$

Step 10.2.4.2

Add $π$ and $2 π$ .

$f (\frac{π}{6}) = \frac{3}{2} \cdot \sin (\frac{3 π}{6})$

Step 10.2.5

Cancel the common factor of $3$ and $6$ .

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Step 10.2.5.1

Factor $3$ out of $3 π$ .

$f (\frac{π}{6}) = \frac{3}{2} \cdot \sin (\frac{3 (π)}{6})$

Step 10.2.5.2

Cancel the common factors.

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Step 10.2.5.2.1

Factor $3$ out of $6$ .

$f (\frac{π}{6}) = \frac{3}{2} \cdot \sin (\frac{3 π}{3 \cdot 2})$

Step 10.2.5.2.2

Cancel the common factor.

$f (\frac{π}{6}) = \frac{3}{2} \cdot \sin (\frac{3 π}{3 \cdot 2})$

Step 10.2.5.2.3

Rewrite the expression.

$f (\frac{π}{6}) = \frac{3}{2} \cdot \sin (\frac{π}{2})$

Step 10.2.6

The exact value of $\sin (\frac{π}{2})$ is $1$ .

$f (\frac{π}{6}) = \frac{3}{2} \cdot 1$

Step 10.2.7

Multiply $\frac{3}{2}$ by $1$ .

$f (\frac{π}{6}) = \frac{3}{2}$

Step 10.2.8

The final answer is $\frac{3}{2}$ .

$y = \frac{3}{2}$

Step 11

Evaluate the second derivative at $x = \frac{7 π}{6}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$- \frac{3 \sin ((\frac{7 π}{6}) + \frac{π}{3})}{2}$

Step 12

Evaluate the second derivative.

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Step 12.1

Simplify the numerator.

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Step 12.1.1

To write $\frac{π}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$- \frac{3 \sin (\frac{7 π}{6} + \frac{π}{3} \cdot \frac{2}{2})}{2}$

Step 12.1.2

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 12.1.2.1

Multiply $\frac{π}{3}$ by $\frac{2}{2}$ .

$- \frac{3 \sin (\frac{7 π}{6} + \frac{π \cdot 2}{3 \cdot 2})}{2}$

Step 12.1.2.2

Multiply $3$ by $2$ .

$- \frac{3 \sin (\frac{7 π}{6} + \frac{π \cdot 2}{6})}{2}$

Step 12.1.3

Combine the numerators over the common denominator.

$- \frac{3 \sin (\frac{7 π + π \cdot 2}{6})}{2}$

Step 12.1.4

Simplify the numerator.

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Step 12.1.4.1

Move $2$ to the left of $π$ .

$- \frac{3 \sin (\frac{7 π + 2 \cdot π}{6})}{2}$

Step 12.1.4.2

Add $7 π$ and $2 π$ .

$- \frac{3 \sin (\frac{9 π}{6})}{2}$

Step 12.1.5

Cancel the common factor of $9$ and $6$ .

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Step 12.1.5.1

Factor $3$ out of $9 π$ .

$- \frac{3 \sin (\frac{3 (3 π)}{6})}{2}$

Step 12.1.5.2

Cancel the common factors.

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Step 12.1.5.2.1

Factor $3$ out of $6$ .

$- \frac{3 \sin (\frac{3 (3 π)}{3 (2)})}{2}$

Step 12.1.5.2.2

Cancel the common factor.

$- \frac{3 \sin (\frac{3 (3 π)}{3 \cdot 2})}{2}$

Step 12.1.5.2.3

Rewrite the expression.

$- \frac{3 \sin (\frac{3 π}{2})}{2}$

Step 12.1.6

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because sine is negative in the fourth quadrant.

$- \frac{3 (- \sin (\frac{π}{2}))}{2}$

Step 12.1.7

The exact value of $\sin (\frac{π}{2})$ is $1$ .

$- \frac{3 (- 1 \cdot 1)}{2}$

Step 12.1.8

Multiply $- 1$ by $1$ .

$- \frac{3 \cdot - 1}{2}$

Step 12.2

Simplify the expression.

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Step 12.2.1

Multiply $3$ by $- 1$ .

$- \frac{- 3}{2}$

Step 12.2.2

Move the negative in front of the fraction.

$- - \frac{3}{2}$

Step 12.3

Multiply $- - \frac{3}{2}$ .

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Step 12.3.1

Multiply $- 1$ by $- 1$ .

$1 (\frac{3}{2})$

Step 12.3.2

Multiply $\frac{3}{2}$ by $1$ .

$\frac{3}{2}$

Step 13

$x = \frac{7 π}{6}$ is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.

$x = \frac{7 π}{6}$ is a local minimum

Step 14

Find the y-value when $x = \frac{7 π}{6}$ .

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Step 14.1

Replace the variable $x$ with $\frac{7 π}{6}$ in the expression.

$f (\frac{7 π}{6}) = \frac{3}{2} \cdot \sin ((\frac{7 π}{6}) + \frac{π}{3})$

Step 14.2

Simplify the result.

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Step 14.2.1

To write $\frac{π}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f (\frac{7 π}{6}) = \frac{3}{2} \cdot \sin (\frac{7 π}{6} + \frac{π}{3} \cdot \frac{2}{2})$

Step 14.2.2

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 14.2.2.1

Multiply $\frac{π}{3}$ by $\frac{2}{2}$ .

$f (\frac{7 π}{6}) = \frac{3}{2} \cdot \sin (\frac{7 π}{6} + \frac{π \cdot 2}{3 \cdot 2})$

Step 14.2.2.2

Multiply $3$ by $2$ .

$f (\frac{7 π}{6}) = \frac{3}{2} \cdot \sin (\frac{7 π}{6} + \frac{π \cdot 2}{6})$

Step 14.2.3

Combine the numerators over the common denominator.

$f (\frac{7 π}{6}) = \frac{3}{2} \cdot \sin (\frac{7 π + π \cdot 2}{6})$

Step 14.2.4

Simplify the numerator.

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Step 14.2.4.1

Move $2$ to the left of $π$ .

$f (\frac{7 π}{6}) = \frac{3}{2} \cdot \sin (\frac{7 π + 2 \cdot π}{6})$

Step 14.2.4.2

Add $7 π$ and $2 π$ .

$f (\frac{7 π}{6}) = \frac{3}{2} \cdot \sin (\frac{9 π}{6})$

Step 14.2.5

Cancel the common factor of $9$ and $6$ .

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Step 14.2.5.1

Factor $3$ out of $9 π$ .

$f (\frac{7 π}{6}) = \frac{3}{2} \cdot \sin (\frac{3 (3 π)}{6})$

Step 14.2.5.2

Cancel the common factors.

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Step 14.2.5.2.1

Factor $3$ out of $6$ .

$f (\frac{7 π}{6}) = \frac{3}{2} \cdot \sin (\frac{3 (3 π)}{3 (2)})$

Step 14.2.5.2.2

Cancel the common factor.

$f (\frac{7 π}{6}) = \frac{3}{2} \cdot \sin (\frac{3 (3 π)}{3 \cdot 2})$

Step 14.2.5.2.3

Rewrite the expression.

$f (\frac{7 π}{6}) = \frac{3}{2} \cdot \sin (\frac{3 π}{2})$

Step 14.2.6

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because sine is negative in the fourth quadrant.

$f (\frac{7 π}{6}) = \frac{3}{2} \cdot (- \sin (\frac{π}{2}))$

Step 14.2.7

The exact value of $\sin (\frac{π}{2})$ is $1$ .

$f (\frac{7 π}{6}) = \frac{3}{2} \cdot (- 1 \cdot 1)$

Step 14.2.8

Multiply $- 1$ by $1$ .

$f (\frac{7 π}{6}) = \frac{3}{2} \cdot - 1$

Step 14.2.9

Multiply $\frac{3}{2} \cdot - 1$ .

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Step 14.2.9.1

Combine $\frac{3}{2}$ and $- 1$ .

$f (\frac{7 π}{6}) = \frac{3 \cdot - 1}{2}$

Step 14.2.9.2

Multiply $3$ by $- 1$ .

$f (\frac{7 π}{6}) = \frac{- 3}{2}$

Step 14.2.10

Move the negative in front of the fraction.

$f (\frac{7 π}{6}) = - \frac{3}{2}$

Step 14.2.11

The final answer is $- \frac{3}{2}$ .

$y = - \frac{3}{2}$

Step 15

These are the local extrema for $f (x) = \frac{3}{2} \cdot \sin (x + \frac{π}{3})$ .

$(\frac{π}{6}, \frac{3}{2})$ is a local maxima

$(\frac{7 π}{6}, - \frac{3}{2})$ is a local minima

Step 16